The Asymptotic Justification of a Nonlocal 1-D Model Arising in Porous Catalyst Theory

Journal of Differential Equations  DE3210 journal of differential equations 134, 68103 (1997) article no. DE963210

The Asymptotic Justification of a Nonlocal 1-D Model Arising in Porous Catalyst Theory* Francisco J. Mancebo and Jose M. Vega E.T.S.I. Aeronauticos, Universidad Politecnica de Madrid, 28040 Madrid, Spain Received April 12, 1996; revised October 1, 1996

An asymptotic model of isothermal catalyst is obtained from a well-known model of porous catalyst for appropriate, realistic limiting values of some nondimensional parameters. In this limit, the original model is a singularly perturbed m-D reaction diffusion system. The asymptotic model consists of an ordinary differential equation coupled with a semilinear parabolic equation on a semi-infinite one-dimensional interval.  1997 Academic Press

1. INTRODUCTION This paper deals with a well-known model of porous catalyst that after suitable nondimensionalization [1, Vol. I] may be written as ut=2u&, 2f (u, v) L

&1

2

vt=2v+;, f (u, v)

in 0,

un=_(1&u)

at 0, (1.1)

in 0,

vn=&(1&v)

at 0,

(1.2)

for t>0, with appropriate initial conditions u=u 0 >0,

v=v 0 >0

in 0,

at t=0.

(1.3)

Here u>0 and v>0 are the reactant concentration and the temperature respectively, 2 is the Laplacian operator, n is the outward unit normal to the smooth boundary of the bounded domain 0/R m (with m1) and the parameters , 2 (Damkohler number), L (Lewis number), ; (Prater number), _, and & (material and thermal Biot numbers) are strictly positive. The nonlinearity f accounts for the reaction rate and is usually of one of the following forms, that are associated with the so-called Arrhenius and LangmuirHinshelwood kinetic laws [1], * This research was supported DGICYT and the EEC Program on Human Capital and Mobility, under Grants PB-94-0416 and CHRX-CT-93-0413. To whom correspondence should be addressed.

68 0022-039697 25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.

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ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

f (u, v)=u p exp(#&#v)

69 (1.4)

f (u, v)=u p(u+k) &q exp(#&#v) p

f (u, v)=u [u+k exp(# a &# a v)]

(1.5) &q

exp(#&#v)

(1.6)

where the reaction orders p and q and the activation energies # and # a are strictly positive. The reaction orders may be non-integers. Porous catalysts usually exhibit a large thermal conductivity and consequently ; is usually very small, the ratio _& is large and & is either small or of order unity, depending on the size of the catalyst (see [1]). In addition, L and , 2 vary in a wide range (from small to large values). Then, the limit ;  0,

_&  

(1.7)

is realistic, and leads to simpler submodels than (1.1)(1.3). If, in addition, & is small and , 2 remains bounded, then the following simpler sub-model is obtained from (1.1)(1.3) ut=2u&, 2f (u, v),

un=_(1&u)

(V 0 &L) dVdt=S 0(1&V)+( ;, 2&)

|

at 0,

f (u, v) dx,

(1.8) (1.9)

0

with appropriate initial conditions, where V 0 and S 0 are the measures of the domain 0 and of its boundary respectively, i.e., V0 =

|

dx, 0

S0 =

|

ds

if m2,

S 0 =2

if m=1, (1.10)

0

and V is the spatial average of the temperature v, i.e., V=V &1 0

|

v(x, t) dx.

(1.11)

0

The model (1.8)(1.9) was obtained in [2] by means of formal, singular perturbation techniques. For a rigorous derivation of a slightly different model (namely, the boundary conditions in (1.1) and (1.8) being replaced by new ones of the Dirichlet type) see [3]. For the rigorous derivation of related simplified sub-models of general reaction-diffusion systems, see [47]. As a by-product of the results below, a fairly direct derivation of (1.8)(1.9) could be readily obtained by means of the ideas in this paper; but for the sake of brevity we shall omit that derivation. The steady states of (1.8)(1.9) and their linear stability were analyzed in [2] for the particular case when f is as given in (1.4) with p=1; some global stability

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MANCEBO AND VEGA

properties for more general, smooth nonlinearities were obtained in [8], and the steady states for some non-Lipschitzian nonlinearities were analyzed [9]. If , 2 is large and & is small then the limit (1.7) is much more interesting because the original model (1.1)(1.2) is singularly perturbed. We shall consider the case when , 2   but ;, is appropriately small. In this limit, the following sub-model of (1.1)(1.3) applies U{= 2U! 2 & f (U, V) u0

as

!  &,

in &
(V 0 , 2(S 0 &L)) dVd{=1&V+(;,&)

|

(1.12) at !=0,

(1.13)

0

f (U, V) d!,

(1.14)

&

with appropriate initial conditions, where V 0 , S 0 and V are given again by (1.10)(1.11) and U is appropriatetely close to u. The new rescaled variables { and ! are {=, 2t,

!=,',

(1.15)

where ' is a co-ordinate along the outward unit normal to 0. Let us now briefly explain (in loose, physical terms, but following the main ideas in the derivation below) where this model comes from. Since , 2 is large, the chemical reaction is very strong and, after some time, the reactant is consumed and u becomes very small in 0 except in a thin boundary layer near the boundary of 0. Since, in addition, ;, and & are small, the temperature v becomes spatially constant (in first approximation) after some time. Finally, if f u >0, after some time, the reactant concentration in the boundary layer depends only on time and on the distance to the boundary of 0 (and not on transversal co-ordinates along 0 if the spatial dimension m is greather than one) in first approximation. Then (1.12)(1.13) gives the evolution of u in the boundary layer, and (1.14) provides the spatially averaged temperature in first approximation. Notice that the sub-model (1.12)(1.14) consists of a 1-D semilinear PDE coupled with an ODE and thus is much simpler than the original model (1.1)(1.2); in particular, the sub-model is independent of the shape of the domain 0 (it depends only on the overall quantities V 0 and S 0 ). A formal derivation of this sub-model, based on singular perturbation techniques, was given in [2], along with the analysis of the steady states, their linear stability and local Hopf bifurcation, for the particular case when the nonlinearity f is as given in (1.4), with p=1. If , 2 is large but & is no longer small, then the temperature does not become spatially constant after some time and a third sub-model is

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71

ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

obtained that consists of the m-D heat equation with appropriate nonlinear boundary conditions, coupled with infinitely many 1-D semilinear equations (one for each point of 0). This non-standard sub-model was derived in [10] via formal, singular perturbation techniques and will be rigorously justified elsewhere [11]. Besides its intrinsic mathematical interest, this sub-model exhibits a large variety of codimension two and three bifurcations that predict interesting dynamic behaviors (see [10]). The main object of this paper is to provide a rigorous derivation of (1.12)(1.14). More precisely, we shall prove that, after some time T, (i) u is very small except in a thin boundary layer and v is spatially constant in first approximation, and (ii) the concentration in the boundary layer and the averaged temperature satisfy (1.12)(1.14), in first approximation, uniformly in tT. Let us now state precisely the assumptions to be made below. We shall consider the limit ,  ,

;,_(,+_)  0,

&0

and

_ &1 =O(1).

(1.16)

The domain 0 and the nonlinearity f will be assumed to be such that (H.1) 0/R m (m1) is a bounded domain, with a connected, C 4+: (for some :>0) boundary if m2. Notice that then 0 satisfies uniformly the interior and exterior sphere conditions: there are two constants, \ 1 >0 and \ 2 >0, such that for every point x of 0, two hyperspheres, of radii \ 1 and \ 2 , S 1 and S 2 , are tangent to 0 at x and satisfy S 1 /0 and S 2 & 0 =[x] (overbars stand for the closure). (H.2) The C 1-function f : [0, [_[0, [  R is such that f (0, v)=0 for all v0 and f (u v)>0 for all u>0 and all v0. (H.3) that

There is a continuous, increasing function, g 1 : [0, [  R such f (u, v)g 1(u)

if u0

and

v0.

(H.4) There are two strictly positive constants, k 1 and k 2 , and a positive, continuous, decreasing function, g 2 : [0, [  R, such that k 2 u f (u, v)k 1 u ug 2(u) f (u, v) (H.5)

if 0u2 if u0

and and

v12, v12,

There are three constants, k 3 >0, k 4 >0 and k 5 >0, such that

k 3  f u(u, v)k 4 ,

| f v(u, v)| k 5 u if 0u_(_+,- k 2 2m) and v1.

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In addition, the initial conditions (1.3) will be assumed to be such that (H.6)

&u 0 & C(0 ) =O(1) and &v 0 & C(0 ) =O(1) in the limit (1.16).

Notice that the non-linearities (1.4)(1.6) satisfy (H.2) for p1 and all (positive) values of the remaining parameters; our results below do not apply (and are not straightforwardly extended) to non-Lipschitzian nonlinearities, such as (1.4)(1.6) if p<1, that are also of practical interest. Assumption (H.4) is satisfied by (1.4)(1.6) only if p=1. Now the restriction is purely technical; if the inequalities in (H.4) are replaced by k 2 u p  f (u, v)k 1 u p and u pg 2  f (u, v) with p>1, then our results below still apply after some (unfortunately, not always obvious) changes, but we do not pursue this extension for the sake of brevity. The first inequality in assumption (H.5) (namely, f u(u, v)k 3 ) is essential in our derivation below; although we have some reasons to believe that the model (1.4)(1.6) should still apply without this restriction, we do not see how to eliminate it completely (we are only able to replace it by k 3 u p&1  f u(u, v) with p>1, but even this small extension requires additional technicalities that are again omitted for the sake of brevity). The remaining restrictions in (H.5) are clearly satisfied by the nonlinearities (1.4)(1.6) for all (positive) values of the parameters. To end up this section let us state the main result of this paper, which is proved at the end of Section 2. Theorem 1.1. Under the assumptions (H.1)(H.6), there are two constants, *>0 and =>0, and for each solution of (1.1)(1.3) there is a solution of Ut= 2U' 2 &, 2f (U, V) U =0

at

'=&,

in &<'<0,

U'=_(1&U )

(V 0 S 0 L) dVdt=&(1&V)+;, 2

|

(1.17) at '=0,

(1.18)

0

f (U, V) d'+(t),

(1.19)

&

and a constant T >0 such that (i) * depends only on the domain 0, = depends only on 0 and on the quantities ,, _, L, ; and &,

(1.20)

and T depends only on 0, on the quantities, (1.20) and on &u 0 & C(0 ) ,

&v 0 & C(0 ) .

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(1.21)

73

ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

(ii)

= and T are such that

==O( ;,_(,+_)), 1

\=+&+ +O(L & ;, ) log 2+ + \ = & +,

T =O(, &1 +L &1 ) log

&1

= , ) log 2+ + _ &(=+&)

\

+

2

+O((&L) &1

(1.22)

in the limit (1.16). (iii)

For all tT we have

|U(&d(x), t)&u(x, t)| [_(=+&)(,+_)] exp[&*,d(x)] if d(x)<\ 1 2,

(1.23)

|V(t)&v(x, t)| =+&

if x # 0,

2

2

|(t)| (=+&) +=, ,

(1.24)

where d(x) is the distance from x to 0 and \ 1 is as defined in assumption (H.1).

2. MATHEMATICAL DERIVATION OF THE APPROXIMATE MODEL Under the assumptions (H.1)(H.2), the parabolic problem (1.1)(1.3) is readily seen to have a unique classical solution in a maximal time interval, 0t0

and

v>0

for all

&u( } , t)& C(0) +&v( } , t)& C(0 )  

(x, t) # 0_[0, T [ , as

tZT

if T<.

(2.1) (2.2)

If, in addition, (H.3) holds then T= and every solution of (1.1)(1.3) is uniformly bounded in 0t<. In order to derive the asymptotic model (1.12)(1.14) we shall first obtain, in Section 2.1, some estimates on related linear elliptic problems and on the solution of (1.1)(1.3). Then, the asymptotic model will be derived in Section 2.2, under the assumptions (H.1)(H.5). Finally, the asymptotic model will be analyzed in Section 3 and some concluding remarks will be drawn in Section 4. In order to avoid too clumsy expresions, we shall only give the orders of magnitude (in the limit (1.16)) of the several constants that appear in this section.

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2.1. Some Preliminary Estimates Let us first prove some results concerning two singularly perturbed, linear elliptic problems, that will be systematically used in the sequel. Lemma 2.1. Let the domain 0/R m be such that assumption (H.1) (at the end of Section 1) holds, and let u and v be the unique solutions of 2u=4 2u

in 0,

un=_(1&u)

at 0,

(2.3)

2v+=4 2u=0

in 0,

vn=&(1&v)

at 0,

(2.4)

where 4, =, _ and & are positive and _>&. As 4  , the following estimates hold [_(_+$ 1 )] exp[&$ 1 d(x)]u(x) [_(_+$ 2 )][cosh($ 2( \ 1 &d 1(x)))cosh($ 2 \ 1 )], _S 0 $ 2 (_+$ 2 )4 2

|

u(x) dx_S 0 $ 1 (_+$ 1 ),

(2.5) (2.6)

0

1
(2.7)

for all x # 0, where \ 1 and S 0 are as defined in assumption (H.1) and Eq. (1.10) respectively, d(x) is the distance from x to 0, d 1(x)=min[d(x), \ 1 ] and the positive constants $ 1 , $ 2 and $ 3 satisfy $ 2 =4- m,

$ 3 ==_$ 1 (_+$ 1 ) &

and

|$ 1 &4| =O(4 &1 )

as 4  , (2.8)

uniformly in =>0, _>0 and &>0. Proof. If the dimension m is equal to 1, then (2.3) and (2.4) are solved in closed-form and (2.5)(2.7) are readily obtained. If m2, let u m = min[u(x): x # 0]>0 and for each x 0 # 0, let S 2 be the outer hypersphere, of radius \2 , that is tangent to 0 at x 0 (assumption (H.1)). If r is the distance to the center of S 2 , let the function w=u m exp[&$ 1(r&\ 2 )], where $ 1 =(m&1)2\ 2 +- (m&1) 24\ 22 +4 2. Then $ 1 satisfies (2.8) and w is such that 2w4 2w

in

0 1 =[x # R m : r>\ 2 ],

w=u m

at 0 1 .

In addition 0/0 1 and wu m u at 0. As a consequence, maximum principles [12] readily imply that uw in 0, and the first inequality (2.5) follows provided that u m _(_+$ 1 ).

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(2.9)

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ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

In order to obtain this inequality, let x 0 be a point where the minimum u m is attained. Then u=w=u m at x 0 and, since uw in 0, we have wnun at x 0 , i.e., $ 1 u m _(1&u m ) and (2.9) follows. Thus the first inequality (2.5) has been obtained. The second inequality (2.5) is obtained in a similar way. Let u M = max[u(x): x # 0 ]; notice that such maximum is attained at 0 because 2u>0 in 0. For each x 0 # 0, let S 1 be the inner hypersphere of radius \ 1 that is tangent to 0 at x 0 (assumption (H.1)), and let the function w be defined as w=u M cosh($ 2 r)cosh($ 2 \ 1 ), where r is the distance to the center of S 1 and the constant $ 2 is as defined in Eq. (2.8). Then 2w4 2w

in

0 1 =[x # R m : r<\ 1 ]/0,

w=u M u

at

0 1 ,

and maximum principles imply that uw in 0 1 . But if x 0 is a point where the maximum u M is attained, then w(x 0 )=u(x 0 ) and wnun at x 0 , i.e., $ 2 u M _(1&u M ), or u M _(_+$ 2 ). Since, in addition, uw for all x 0 # 0, the second inequality (2.7) follows when d(x)\ 1 . In order to prove that this inequality also holds when x # 0 2 =[x # 0 : d(x)>\ 1 ], notice that if 0 2 {<, then the maximum of u in 0 2 is attained at 0 2 (because 2u>0 in 0 2 ) and u(_(_+$ 2 )) cosh($ 2 \ 1 ) at 0 2 . Thus the second inequality (2.5) has been obtained. In order to prove that (2.6) holds integrate Eq. (2.3) in 0, integrate by parts and take into account the boundary condition to obtain 4 2  0 u dx=_  0 (1&u), and apply (2.5). Finally, the first inequality (2.7) is readily obtained via maximum principles when taking into account that =4 2u>0 in 0. In order to obtain the second inequality (2.7), notice that the function v 1 =v&=(1&u) satisfies 2v 1 =0

in 0,

v 1 n=&(1&v 1 )+=(_&&)(1&u),

and, since u_(_+$ 1 ) at 0 (see (2.9)) and _>&, maximum principles readily imply that v 1 1+=(_&&) $ 1 (_+$ 1 ) & in 0, or v1+=_$ 1  (_+$ 1 ) & in 0. Thus, the proof is complete. Let us now prove some estimates on the solution of (1.1)(1.3). In particular we show that, after some time, u becomes quite small except in a boundary layer near 0 (Lemma 2.2) and |v&V| also becomes quite small (Lemma 2.3), where V is the spatial average of v. Lemma 2.2. Under the assumptions (H.1)(H.4) and (H.6) (at the end of Section 1) there is a constant T, depending only on &u 0 & C(0 ) ,

&v 0 & C(0 ) ,

,, _, L, ;, and &,

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(2.10)

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MANCEBO AND VEGA

and satisfying T=O(, &2 log(2+,_+_,)+, &1 +(&L) &1 log(2+;, 2&))

(2.11)

in the limit (1.16), such that every solution of (1.1)(1.3) satisfies u 1
12
in 0 if tT,

(2.12)

where u 1 , u 2 and v 1 are the unique solutions of 2u 1 =2k 1 , 2u 1

in 0,

u 1 n=_(1&u 1 )

at 0,

(2.13)

2u 2 =k 2 , 2u 2 2

in 0,

u 2 n=_(1&u 2 )

at 0,

(2.14)

2v 1 +;k 1 , 2u 2 =0

in 0,

v 1 n=&(1&v 1 )

at 0,

(2.15)

with the constants k 1 >0 and k 2 >0 as defined in assumption (H.4). Proof.

Let : 1 >0 be the smallest eigenvalue of 2. 1 +: 1 . 1 =0

in 0,

. 1 n+&. 1 =0

at 0,

(2.16)

and let . 1 be the associated eigenfunction such that in 0,

. 1 >0

max[. 1(x): x # 0 ]=1.

(2.17)

. 1 and : 1 are readily seen to satisfy min[. 1(x) : x # 0 ]  1

and

: 1 =S 0 &V 0 +o(& 2 )

as &  0, (2.18)

where S 0 and V 0 are as defined in (1.10). The proof proceeds in five steps. Step 1. u and v are such that u
and

v
if

x # 0 and

t0,

(2.19)

where the constants A 1 and A 2 depend only on the quantities (2.10) and satisfy 1
and 1
in the limit (1.16).

(2.20)

Let A 1 =1+max[u 0(x) : x # 0 ], that satisfies (2.20) according to assumption (H.6). The function w, defined as w=A 1 &u, is readily seen to satisfy w>0 in 0 if t=0, wt&2w>0 in 0 if t>0 and wn+_w>0 in 0 if t>0 and, consequently, maximum principles [12] imply that w>0 in 0 if t>0. Then, the first inequality (2.19) follows.

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77

ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

In order to obtain the second inequality (2.19), let the function v 2 be defined as the unique solution of the linear problem 2v 2 +;, 2g 1(A 1 )=0

in 0,

v 2 n+&(v 2 &1)=0

at 0,

where the function g 1 is as defined in assumption (H.3). Maximum principles readily imply that v 2 1+;, 2g 1(A 1 ) . 1 [: 1 min[. 1(x) : x # 0 ]]

in 0,

where : 1 and . 1 are as defined above, or, according to (2.17)(2.18) and the first estimate (2.20), max[v 2(x) : x # 0 ]=O(1+;, 2&)

in the limit (1.16).

(2.21)

On the other hand, the function w, defined as w=max[v 0(x): x # 0 ]+ v 2 &v satisfies (see assumption (H.3)) w>0 in 0, wn+&w0

if t=0, at 0,

L &1 wt>2w in 0,

if t>0,

if t>0,

(2.22) (2.23)

and consequently maximum principles imply that w0 in 0 if t0. Then assumption (H.6) and Eq. (2.21) yield the second inequality (2.19), with A 2 satisfying (2.20), and the step is complete. Step 2. There is a constant T 1 , depending only on the quantities (2.10), such that T 1 =O(1&L) in the limit (1.16), and v12

in 0

if

tT 1 .

(2.24)

Let the constant A 3 12 be such that A 3 . 1 >1&v 0 in 0. Notice that A 3 may be chosen to be bounded, according to assumption (H.6) and Eq. (2.18). The function w defined as w=v&1+A 3 . 1 exp(&: 1 Lt) is readily seen to satisfy (2.22) and (2.23) and, as above, maximum principles imply that w0 in 0 if t0. Then (2.24) holds with T 1 =(: 1 L) &1 log (2A 3 ) and the result follows. Step 3. There is a constnat T 2 , depending only on the quantities (2.10), such that 0
in 0

if

tT 2 .

Let the constant A 4 >0 be defined as A 4 = g 2(A 1 ). According to the assumption (H.2) and the results in steps 1 and 2, f (u, v)>A 4 u in 0 for

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MANCEBO AND VEGA

all tT 1 , and the function w=A 1 exp[ &A 4 , 2(t&T 1 )]+1&u satisfies w>0

in 0

if t=0,

wn+_w0

wt>2w&A 4 , 2w

at 0,

in 0

if

tT 1 , (2.25)

if t>T 1 .

(2.26)

As a consequence, maximum principles imply that w0 if x # 0 and tT 1 , and the result follows with T 2 =T 1 +(A 4 , 2 ) &1 log A 1 . Step 4. There is a constant T 3 , depending only on the quantities (2.10), such that 0
in 0

if tT 3 ,

(2.27)

where u 1 and u 2 are as given by (2.13)(2.14). For the sake of brevity we shall only obtain the second inequality (2.27); the first inequality is obtained in a completely similar way. Let the function u 3 be the unique solution of the linear problem 2u 3 =k 2 , 2u 3

in 0,

u 3 n=_(1&u 3 )

at 0,

and let the constant $ be defined as $=min[u 3(x): x # 0 ]2. According to Lemma 2.1, $ is such that $[_2(_+$ 1 )] exp(&$ 1 D2),

(2.28)

where D is the diameter of the domain 0 and |$ 1 &- k 2 ,| =O(, &1 ). Then the function w=(1+$) u 1 &u 3 &$ is such that 2w
(2.29)

Now, if the constant A 5 >0 is such that A 5 >u( } , T 2 )&u 3 in 0 (A 5 may be chosen to be such A 5 2, according to the result in step 3), then the function w=u 3 &u+A 5 exp[&, 2k 2(t&T 2 )] satisfies (2.25)(2.26) with A 4 and T 1 replaced by k 1 and T 2 respectively, and again maximum principles imply that w0 in 0 if tT 2 . Then the second inequality (2.27) holds provided that T 3 &T 2 =(, 2k 2 ) &1 |log[A 5(,+- 2mk 2 _)$,]| and, when taking into account (2.28), the result follows.

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ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

79

Step 5. There is a constant T, depending only on the quantities (2.10), such that 0T&T 3 =O(1&L) log(2+;, 2&) in the limit (1.16), and in 0

v1+v 1

if

tT.

(2.30)

According to the results in steps 14, if tT 3 then uu 2 ( 1) and 12vA 2 , and according to assumption (H.4), f (u, v)k 1 u in 0. Let the function w be defined as w=v 1 &v+A 6 . 1 exp[ &: 1 L(t&T 3 )], where A 6 1 satisfies A 6 . 1 A 2 in 0; notice that, according to (2.18), A 6 may be chosen such that A 6 =O(A 2 )=O(1+;, 2&).

(2.31)

Also the function w satisfies (2.22)(2.23) with t replaced by t&T 3 , and maximum principles imply that w0 in 0 for all tT 3 . As a consequence, (2.30) holds with T=T 3 +(: 1 L) &1 log(1+A 6 ). Finally, T&T 3 =O(1&L) log(2+;, 2&) in the limit (1.16), as obtained when taking into account (2.18) and (2.31). Thus the step and the proof of the Lemma are complete. Lemma 2.3. Under the assumptions of Lemma 2.2 there are two constants, +>0 and T $T such that (i) + and T $&T depend only on (the domain 0 and) ,, _, L, ; and &,

(2.32)

(ii) +=O(&+;,_(,+_)) and T $&T=O(L &1 ) (_+,)&))+] in the limit (1.16); and (iii) if tT $ then |v&V| +

log[(1+(;_,

for all x # 0,

(2.33)

where V(t) is the spatial average of v, i.e., V(t)=V &1 0 Proof.

|

v(x, t) dx,

with

0

V0=

|

dx.

(2.34)

0

Let us define the new time variable {=Lt&T.

(2.35)

Then the spatial average of v satisfies

_

dVd{= &

|

0

(1&v) ds+;, 2

|

f (u, v) dx 0

&
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0

(2.36)

80

MANCEBO AND VEGA

as obtained upon integration of (2.2) in 0, integration by parts, substitution of the boundary condition and multiplication by V &1 0 . If (2.36) is substracted from (2.2) then we obtain (v&V){=2(v&V)+;, 2f (u, v)

_

|

& &V &1 0 (v&V)n=&(1&v)

(1&v) ds+;, 2 0

|

&

f (u, v) dx , 0

at 0.

(2.37) (2.38)

On the other hand, according to assumption (H.4) (at the end of Section 1) and Lemmas 2.1 and 2.2, we have in 0

f (u, v)k 1 u 2

if {0,

0
in 0,

(2.39)

&1&u 2 & C(0) =O(,(,+_)),

|

u 2(x) dx=O(_,(_+,)),

&v 1 & C(0 ) =O(1+;,_(,+_)&)

(2.40)

0

in the limit (1.16), where u 2 and v 1 are as given by (2.14) and (2.15). The proof proceeds in three steps. Step 1. The following inequality holds

|

(v&V) 2 dxB 2 exp(&2# 1 {)+2# 1 B 1

0

|

{

_

&v( } , !)&V(!)& C(0 ) exp[2# 1(!&{)] d!

(2.41)

0

if {0, where # 1 >0 depends only on the domain 0, B 1 >0 and B 2 >0 depend only on (the domain 0 and) the quantities (2.32), and B 1 = O(&+;_,(_+,)) and B 2 =O(1+;_,(_+,)&) 2 in the limit (1.16). If (2.37) is multiplied by v&V, the resulting equation is integrated in 0, integration by parts is applied and (2.38) is substituted, then the following equation results 1 d 2 d{

|

0

|

(v&V) 2 dx=&

|{v| 2 dx+;, 2

0

+&

|

(v&V) f (u, v) dx

0

|

(v&V)(1&v) ds,

0

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(2.42)

81

ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

where we have taken into account that, according to (2.34)

|

(v&V) dx=0

for all {0.

(2.43)

0

Now we take into account the following PoincareFriedrichs-type property. There is a constant #>0, depending only on the domain 0, such that for every . # W 12(0) such that  0 .(x) dx=0, the following inequality holds (see, e.g., [13, p. 45, Eq. (2.12)]) #  0 .(x) 2 dx 0 |{.| 2 dx. If, in addition, (2.12), (2.39), and (2.40) are taken into account, then the following inequality results from (2.42) 1 d 2 d{

|

(v&V) 2 0

 &#

|

(v&V) 2 dx+B 1 &v( } , {)&V({)& C(0)

if {0

(2.44)

0

where B 1 =;, 2k 1  0 u 2(x) dx+&  0 |v 1(s)&1| ds=O(&+;_,(_+,)) in the limit (1.16) (see (2.40)), as stated. And we only need to apply Gronwall's lemma to obtain (2.41) with B 2 = 0 v 1(x) 2 dx 0 [v(x, 0)&V(0)] 2 dx (see (2.12) and (2.32)). According to (2.40), B 2 =O(1+;_,(_+,)&) 2 in the limit (2.16), and the step is complete. Step 2. The following inequality holds &v&V& C(0 )_[{, {+1] # 2 [B 3 +&v&V& L2(0_]{&1, {+1[ ]

(2.45)

for all {>1, where # 2 >0 depends only on the domain 0 and B 3 >0 depends on (0 and) the quantities (2.32), and satisfies B 3 =O(&+;_,(_+,)) in the limit (1.16). In order to obtain (2.45) we decompose v&V as v&V=&&V.+w 1 +w 2 ,

(2.46)

where . and w 1 are uniquely defined by the linear problems 2.=.

in 0,

.n+&.=1

w 1 {=2w 1 &w 1 +;, 2f (u, v) w 1 n+&w 1 =0 at 0,

if

at 0,

in 0, {0,

if

w 1 =0

{>0, in 0,

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(2.47) if

{=0,

82

MANCEBO AND VEGA

and w 2 satisfies, for all {0, w 2 {&2w 2 =&&V.+w 1 +(&.&1) V &1 0

_

_ ;, 2 w 2 n+&w 2 =&

|

f (u, v) dx+&

0

|

(1&v) ds 0

&

in 0,

(2.48)

at 0.

Now, . and w 1 are such that .>0

in 0,

&.& C(0 ) =O(1)

0w 1 (2k 2 k 1 ) ;(1+B 4 .)

in 0

as &  0, if

{0,

(2.49) (2.50)

where k 1 and k 2 and u 2 are as defined in assumption (H.4) (at the end of Section 1) and Lemma 2.2, and B 4 is given B 4 =_ max[1&u 2(x): x # 0] min[.(x) : x # 0] and satisfies (see (2.39)) B 4 =O(,_(,+_)),

in the limit (1.16).

(2.51)

In order to obtain (2.49) we only need to apply maximum principles to (2.47), and take into account that &  0 is a regular limit of (2.47). Similarly, (2.50) follows when taking into account that if either w=w 1 or w=(2k 1 k 2 );(1&u 2 +B 4 .)&w 1 then w{&2w+w0 in 0, wn+ &w0 at 0 if {0, and w0 in 0 if {=0, and applying maximum principles. Finally, (2.34), (2.39)(2.40), and (2.49)(2.51) imply that the sup norm of the right side of (2.48) is bounded above by a constant B 5 >0 that depends only on the quantities (2.32) and satisfies B 5 =O(&+;_,(,+_)),

in the limit (1.16).

(2.52)

Then, local parabolic estimates [14, p. 355] readily imply that for each p2 there is a constant $ p , depending only on the domain 0, such that &w 2 & Wp2, 1 (0_] {, {+1[) $ p(B 5 +&+&w 2 & L2(0_] {&1, {+1[ ) ) for all {2. If p2 is taken such that p>(m+2)2 (m=dimension of 0), then imbedding theorems [14, p. 80] imply that there is a constant $$p , depending only on 0, such that &w 2 & C(0 )_[{, {+1] $$p &w 2 & Wp2, 1(0_] {, {+1[ ) . These two inequalities and (2.46), (2.49)(2.52) readily imply the stated result, and the step is complete.

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83

ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

Step 3. The result in the statement of this lemma holds. For each positive integer k, let P k 0 and Q k 0 be defined as P k =&v&V& C(0_[k, k+1]) ,

Q k =&v&V& L2(0_]k, k+1[ ) .

(2.53)

Then, according to (2.12), (2.34) and (2.40), we have P 0 +P 1 B 6 ,

(2.54)

where B 6 depends only on the quantities (2.32) and satisfies B 6 =O(1+;,_(,+_) &),

in the limit (1.16),

(2.55)

and, according to the results in steps 1 and 2, Q 2k 

|

k+1 k

_

B 2 +2# 1 B 1

|

k&1

&v( } , !)&V(!)& C(0 ) exp(2# 1 !) d! 0

&

_exp(&2# 1 {) d{

(2.56)

_

k

&

(2# 1 ) &1 exp(2# 1 ) B 2 +B 1 : P q exp(2# 1 q) exp(&2# 1 k), q=0

P k # 2(B 3 +Q k&1 +Q k ),

(2.56) (2.57)

for all k1. Also, if (2.57) is substituted into (2.56) and (2.54) is taken into account, then we obtain

\

k&1

+

Q 2k B 7 Q k + 2B 7 : Q s exp(2# 1 s)+B 8 exp(&2# 1 k) s=1

for all

k2, (2.58)

where the constants B 7 and B 8 depend only on the quantities (2.32) and satisfy B 7 =O(&+;_,(_+,)),

B 8 =O(1+;_,(_+,)&) 2,

in the limit (1.16). (2.59)

The inequalities (2.54) and (2.58) imply that Q k 5B 7 +(1+B 6 +B 12 8 ) exp[&# 1(k&1)]

for all

k1,

as readily seen by means of an induction argument. Then we only need to take into account (2.35), (2.53), (2.55), (2.57) and (2.59) to complete this step and the proof of the lemma.

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84

MANCEBO AND VEGA

2.2. Derivation of the Asymptotic Model Let us define the function U as the unique solution of the following semilinear parabolic problem Ut=2U&, 2f (U, V) Un=_(1&U)

in 0

at 0,

if t>0,

if t>0;

(2.60) in 0,

u=u 0

if t=0, (2.61)

where V is the spatially averaged temperature, defined in (2.34), that satisfies (see (2.35)(2.36)) (V 0 L)(dVdt)=&

|

(1&v) ds+;, 2

0

|

f (u, v) dx.

(2.62)

0

In order to derive the asymptotic model (1.8)(1.9) we shall first prove that there is a constant T " such that if tT " then the following properties hold: (i) v>1 in 0, V>1 and U is very small except in a boundary layer near the boundary of 0 (Lemma 2.4); (ii) |u&U| is appropriately small in the boundary layer (Lemma 2.5); and (iii) |{ U | is appropriately small in the boundary layer, where { is the spatial gradient along the hypersurfaces parallel to 0 (Lemma 2.8). Then the asymptotic model will be obtained in Theorem 1.1 as follows. As a consequence of property (iii), U depends only on the distance to 0 in first approximation and thus U satisfies a 1-D parabolic equation in first approximation. In addition, since |v&V| is appropriately small (Lemma 2.3), u and v can be replaced by U and V in Eq. (2.62) in first approximation, and the model (1.12)(1.14) follows. Let us begin with property (i). Lemma 2.4. Under the assumptions (H.1)(H.4) and (H.6) (at the end of Section 1) there is a constant T "1 2T $ such that T "1 &2T $ depends only on the quantities (2.32) and satisfies T "1 &2T $=O(&L) &1 log[2+&(,+_) ;,_] in the limit (1.16), and V>1,

v>1

and

u 1 Uu 2

in 0,

for all

tT "1 ,

where u 1 , u 2 and T $ are as defined in Lemmas 2.2 and 2.3. Proof.

According to the results in Lemmas 2.12.3, if tT $ then ;, 2

|

f (u, v) dx;k 2 S 0 _,2[,+_ - m2k 1 ],

0

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(2.63)

85

ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

where S 0 = 0 1ds. As a consequence V is such that (see also (2.33)(2.34) and (2.62)) (V 0 L)(dVdt)&S 0(1&V&+)+;S 0 k 2 _,2[,+_- m2k 1 ], where + is as defined in Lemma 2.3. Then we only need to apply Gronwall's lemma and take into account that &  0 in the limit (1.16) and Eq. (2.33) to obtain the first two inequalities (2.63) for tT "1 , with T"1 as stated. Finally, the last two inequalities (2.63) readily follow by the argument in the proof of Lemma 2.2, step 4. Thus the proof is complete. Now, we show that |u&U| is conveniently small. Lemma 2.5. Under the assumptions (H.1)(H.6) (at the end of Section 1), let + and T"1 be as defined in Lemmas 2.3 and 2.4. Then there is a constant T"2 T"1 such that T "2 &T "1 depends only on the quantities (2.32) and satisfies T"2 &T"1 =O(, &1 ) log(1+1+) in the limit (1.16), and in 0,

|U&u| 4k 5 +u 2 k 3

for all tT "2 ,

(2.64)

where k 3 , k 5 and u 2 , are as defined in assumption (H.5) and Lemma 2.2. Proof. Since, according to Lemma 2.1, u 2 _(_+, - k 2 2m), when using assumption (H.5) (at the end of Section 1) and the results in Lemmas 2.22.4, and applying the mean function theorem, we have f (u, v)& f (U, v)=h(x, t)(u&U),

with

h(x, t)k 3 ,

| f (U v)& f(U, V)| k 5 +u 2 in 0, for all tT"1 . As a consequence, the functions w \ =2k 5 +u 2 k 3 +[_(_+,- k 2 2m)] exp[&k 3 , 2(t&T"1 )2]\(u&U) are readily seen to satisfy w \ t&2w \ +, 2h(x, t) w \ 0 w \ n+_w \ >0 w \ =0

in 0

at 0, if

in 0,

if tT"1 , t=T "1 ,

and maximum principles imply that w \ 0 in 0 for all tT "1 . Since, in addition, u 2 [_(_+$ 1 )] exp(&$ 1 D2), where D is the diameter of the domain 0 and |$ 1 &- k 2 2 ,| =O(, &1 ) as ,   (Lemma 2.1), the result follows. Thus, the proof is complete.

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86

MANCEBO AND VEGA

The following result gives a bound on the spatial derivatives of the solution of (2.60)(2.61). Lemma 2.6. Under the assumptions of Lemma 2.4, there is a constant + 1 >0, depending only on the quantities (2.32), such that + 1 =O(_,(,+_)) in the limit (1.16) and |{U(x, t)| + 1 exp[&,d 1(x) - k 2 2m]

if

x # 0

tT"1 +2, 2,

and

(2.65)

where d 1(x)=min[ \ 1 , d(x)], d(x) is the distance from x to 0 and \1 , k 1 and T"1 are as defined in assumptions (H.1) and (H.4) (at the end of Section 1) and in Lemma 2.4. Proof. According to assumption (H.4) and the results in Lemmas 2.1 and 2.4, we have 0 f (U, V)k 1 U,

(2.66)

0U[2_(_+, - k 2 2m)] exp[&,d 1(x) - k 2 2m], 01&U[1+_d(x)] $ 1 (_+$ 1 ),

(2.67)

in 0, if tT"1 , where $ 1 depends only on the quantities (2.32) and satisfies |$ 1 &- 2k 1 ,| =O(1,) in the limit (1.16). In order to bound |{U(x 0 , t 0 )|, with x 0 # 0 and t 0 T "1 +2, 2, we shall distinguish three cases, depending on the relative values of d(x 0 ), 1, and 1_. Case 1. d(x 0 )min[1_, 1,]. In this case we introduce the new variables w, ! and { defined as w=1&U,

x=x 0 +=!

and

t=t 0 += 2{,

(2.68)

where ==min[1_, 1,], to rewrite (2.60)(2.61) as w{=2 ! w+(=,) 2 f (U, V)

in 0 = ,

wn+=_w=0 at 0 = . (2.69)

Now if B and B$ are the balls with center at the origin and radii 1 and 2 respectively, local L p estimates up to the boundary [14, p. 355] and imbedding theorems [14, p. 80] imply that there is a fixed constant K such that &{ ! w& C(B=_[0, 1]) K[(=,) 2 & f (U, V)& Lp(B$=_]&1, 1[ ) +&w& L1(B$=_]&1, 1[ ) ], where B = =B & 0 = , B$= =B$ & 0 = and p=(m+3)2. Notice that although 0 = & B$ and the coefficient =_ in the boundary condition depend on =, the

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87

ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

constant K may be chosen to be independt of = because = is bounded above (see (1.16)) and, as =  0, 0 = & B$ converges to a part of a hyperplane and =_ remains bounded above. Now, when using (2.66) and (2.67) and taking into account that =,1 we obtain &{ ! w& C(B=_[0, 1]) 2K[(2# m ) 1p k 1 =,_(_+,- k 2 2m)+2# m(1+2=_) $ 1 (_+$ 1 )], where # m is the measure of the unit ball of R m, or when coming back to the original variables, |{U(x 0 , t 0 )| 2K[(2# m ) 1p k 1 ,_(_+,- k 2 2m)+2# m(max[,, _]+2_) $ 1 (_+$ 1 )]. (2.70) Case 2. _>, and 1_
(2.71)

Case 3. d(x 0 )>1,. Now we take ==1, and use the variables ! and { defined in (2.68) to rewrite (2.60) as U{=2U& f (U, V)

in 0 = .

Since B$ & 0 = is void, local interior estimates and imbedding theorems yield &{ ! U& C(B_[0, 1]) K[& f (U, V)& L p(B$_]&1, 1[ ) +&U& L1(B$_]&1, 1[ ) ],

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88

MANCEBO AND VEGA

where again K is a fixed constant and p=(m+3)2, or when using (2.67) &{ ! U& C(B_[0, 1]) 2K(2# m ) 1p k 1[_(_+,- k 2 2m)] exp[&,(d 1(x 0 )&=) - k 2 2m]. Then we only need to come back to the original variables to obtain |{U(x 0 , t 0 )| 2K(2# m ) 1p k 1[,_(_+,- k 2 2m)] _exp[&,(d 1(x 0 )&1,)- k 2 2m],

(2.72)

where we have taken into account that d 1(x 0 &=)d 1(x 0 )&==d 1(x 0 )&1,. Finally, since one of the cases 13 above necessarily holds, Eqs. (2.70) (2.72) yield the stated result, and the proof is complete. In order to bound the gradient of U along the hypersurfaces parallel to 0 (i.e., orthogonal to the normals to 0 at each point) we first collect some facts from differential geometry. Let 0 1 be defined as 0 1 =[x # 0 : d(x)<\ 1 2]

(2.73)

where \ 1 is defined as in assumption (H.1) (at the end of Section 1) and, as above, d(x) is the distance from x to 0. According to assumption (H.1), the hypersurfaces parallel to 0 are of class C 4, and simply cover 0 1 . Notice that if x=x 0(' 2, ..., ' m ) is a C 4-regular parametric representation of a part of one of these hypersurfaces, H, and n=n(' 2, ..., ' m ) is the outward unit normal to H, then x=' 1n(' 2, ..., ' m )+x 0(' 2, ..., ' m )

(2.74)

defines a local C 3-coordinate system of R m such that the hypersurfaces ' 1 = constant are precisely those parallel to H (and to 0). Also, the covariant components of the metric tensor associated with these co-ordinate system are such that g 11 =n } n#1

and

g 1k =n } (' 1n ' k +x 0' k )#0

if k{1,

(2.75)

where the dot stands for the inner product of R m. Then the contravariant components of the metric tensor satisfy g 11 #1,

g 1k #0

if k{1.

With these facts in mind we can prove the following result.

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(2.76)

ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

89

Lemma 2.7. Let t~ 0 be a unit vector that is tangent to a hypersurface, H, parallel to 0, at p # 0 1 . Then there are a neighborhood N of p in R m, a C 3-vector field t~: N  R 3, two vectors a 1 and a 2 and two scalars, b 1 and b 2 , such that the following properties hold: a 1 , a 2 , b 1 and b 2 depend continuously on p and t~ 0 . (ii) t~ =t~ 0 at p, t~ } t~ =1 in N and, for each q # N & 0 1 , t~(q) is tangent to the hypersurface parallel to 0 passing through q . (iii) If I/R is an open interval and U : (N & 01 )_I  R is a C 3, 1-function satisfying (i)

Ut=2U+.

in

(N & 0 1 )_I

(2.77)

then the C 2, 1-function w={U } t~ satisfies wt=2w+a 1 } {w+a 2 } {U+b 1 w+{. } t~ at p, wn={(Un) } t~ +b 2 w at p,

for all

for all t # I,

t # I,

(2.78) (2.79)

where n is the outward unit normal to H at p. Proof. Let H be the hypersurface parallel to 0 passing through p and let x=x 0(' 2, ..., ' m ) be a C 3-parametric representation of H in a neighborhood of p, where (' 2, ..., ' m ) are Fermi geodesic coordinates defined as follows. The first coordinate ' 2 is an arclength along the geodesic of H, C, that is tangent to t 0 at p. If m>2, then the remaining coordinates are arclengths along m&2 geodesics that are tangent at each point of C to e 3 , ..., e m , where [e 2 , ..., e m ] is an orthonormal frame moved along C by parallelism on H, such that e 2 =t~ 0 at p. In addition, (' 2, . . ., ' m ) are chosen with origin at p and such that the line ' 3 = } } } =' m =0 is the geodesic C. That coordinate system is well defined in a neighborhood of p and such that x 0' i } x 0' j =$ ij

if ' 3 = } } } =' m =0,

x 0'2 =t~ 0

at p,

(2.80)

where $ ij is the Kronecker symbol (see, e.g., [15, p. 335]). Then (2.74) defines a C 3-coordinate system of R m in a neighborhood N of p, whose metric tensor satisfies (see (2.75)(2.76) and (2.80)) g ij = g ij =$ ij

for all

i, j=1, ..., m,

if ' 1 =' 3 = } } } =' m =0.

Also, (2.77) may be written as m U = : t i, j=1

_

g ij

U  2U  +G &12 i (G 12g ij ) j +., i j ' ' ' '

&

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(2.81)

90

MANCEBO AND VEGA

where G is the determinant of the m_m matrix (g ij ). If this equation is derivated with respect to ' 2 and it is taken into account that, according to (2.81), g ij' 2 =G' 2 =0 at p, then the following equation results w 1 t=2w 1 +A } {U+{. } t~ 0 ,

w 1 n={(Un) } t~ 0

at p,

where w 1 =U' 2 #{U } (' 1n '2 +x 0'2 ) and A is the vector field whose contravariant components are m

A j= : i=1

  G &12 i (G 12g ij ) , 2 ' '

_

&

(2.82)

and w={U } (' 1n ' 2 +x 0' 2 )- g 22 #w 1 - g 22 is readily seen to satisfy (2.78) and (2.79) at p with a 1 =2( g 22( p)) &12 {(- g 22( p)), b 1 =( g 22( p)) &12 2(- g 22( p)),

a 2 =( g 22( p)) &12 A,

b 2 = &( g 22( p)) &12  - g 22( p)n

(2.83)

and t~ =(' 1n ' 2 +x 0' 2 )- g 22 . Notice that t~ is a unit vector tangent to the hypersurfaces ' 1 =constant (that are parallel to H) along the parametric lines associated with the coordinate ' 2. Finally, a 1 , a 2 , b 1 and b 2 depend continuously on p and t~ 0 (see (2.82)(2.83) and take into account that g ij , g ij and their first and second order derivatives depend continuously on p and t~ 0 ). Thus the proof is complete. A bound to the gradient of U along the hypersurfaces parallel to 0 is given in the following result Lemma 2.8. Under the assumptions (H.1)(H.6) (at the end of Section 1), let \ 1 , k 1 , k 3 , T "1 and + 1 be as defined in assumptions (H.1), (H.4), and (H.5) and in Lemmas 2.4 and 2.6. Then there are two constants, T "T "1 +1, 2 and + 2 >0, depending only on the quantities (2.32) such that T "&T "1 = O(, &2 ) log , and + 2 =O(_(,+_),) in the limit (1.16), and |{ U| + 2 exp[&- k ,d(x)]

if x # 0 1 ,

and

tT",

(2.84)

where U is a solution of (2.60)(2.61), { U is the gradient of U along the hyper-surfaces parallel to 0, k=min[k 2 4m, k 3 3]>0,

(2.85)

0 1 is as defined in (2.73) and, as above, d(x) is the distance from x to 0.

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ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

Proof.

91

Let us consider the function

w 1 =[+ 2 2++ 1 exp(&k, 2(t&1, 2 &T "1 ))] exp(&- k ,d(x)),

(2.86)

where + 1 >0 is as defined in Lemma 2.6 and + 2 >0 is to be defined below (see Eq. (2.88)). When taking into account that 0 1 may be covered by a finite number of coordinate systems such as that in (2.74), with ' 1 =&d(x), it is readily seen that 2w 1 (k, 2 +k 6 ,) w 1 , w 1 n=- k ,w 1

|{w 1 | =- k ,w 1 at 0,

in 0 1 ,

(2.87)

for all t,

where k 6 =- k max[ |G ' 1 G| in 0 1 ]. Let the continuous functions a 1 , a 2 : 0 1  R m and b 1 , b 2 : 0 1  R be as defined in Lemma 2.7, and let k 7 >0 be a common upper bound of |a 1 |, |a 2 |, and |b 1 | in 0 1 , and of |b 2 | on 0. Notice that k 7 depends only on 0. If ,>max[k 7 - k, 6(1+k 6 +k 7 ) - kk 3 , - 6k 7 k 3 ] (recall that ,   in the limit (1.16)) and + 2 is defined as + 2 =(4+ 1 , 2 ) max[k 7 , 2[k 3 , 23&(k 6 +k 7 ) - k ,&k 7 ], , 2 exp( &- k ,\ 1 4)]

(2.88)

then + 2 >0 and w 1 satisfies w 1 t>2w 1 +a 1 } {w 1 +a 2 } {U+b 1w 1 &k 3 , 2w 1 at p,

for all p # 0 1 , (2.89)

w 1 > |{U|

if d(x)=\ 1 2,

w 1 n&b 2 w 1 >0 at p,

(2.90)

for all p # 0,

(2.91)

provided that tT "1 +1, 2, as readily seen when taking into account (2.65) and (2.85)(2.87). Now, if we show that |{ U| w 1

in 0 1

for all tT"1 +1, 2

(2.92)

then (2.84) follows with + 2 as given by (2.88) and T"=T"1 +1, 2 + (k, 2 ) &1 log(1++ 1 + 2 ). In order to show that (2.92) holds first notice that, since |{ U|  |{U|, according to (2.86) and the result in Lemma 2.6,

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MANCEBO AND VEGA

w 1 > |{ U| in 0 1 if t=1, 2 +T "1 . Assume for contradiction that there is a first value of t, T 0 , and a point p # 0 1 such that |{ U( p, T 0 )| =w 1 ,

in 0 1

|{ U| w 1

if tT 0 .

(2.93)

Let H be the hypersurface parallel to 0 at p. According to the definition of { , there is a unit vector, t~ 0 , that is tangent to H at p and such that |{ U( p, T 0 )| ={ U( p, T 0 ) } t~ 0 ={U( p, T 0 ) } t~ 0 , and if t~ and N are as given in Lemma 2.7, then {U } t~ ={ U } t~  |{ U|

in 0 1 & N,

for all t.

(2.94)

As a consequence, w={U } t~ satisfies (see (2.93)) w=w 1

at (x, t)=( p, T 0 ),

ww 1

in 0 1 & N

if tT 0 .

(2.95)

Also, the result (iii) in Lemma 2.7 implies that wt2w+a 1 } {w+a 2 } {U+b 1 w&, 2k 3 w at

(x, t)=( p, T 0 )

wn=&_w+b 2 wb 2 w

if p # 0 1 , at

(x, t)=( p, T 0 )

(2.96) if p # 0, (2.97)

where we have taken into account (2.60)(2.61) and that, according to Lemmas 2.1 and 2.4 and to assumption (H.5) (at the end of Section 1), fU (U, V)>k 3 and w=w 1 >0 at (x, t)=( p, T 0 ). In order to get contradiction, we shall distinguish three cases: (i) If p # 0 1 then w 2 =w 1 &w satisfies {w 2( p, T 0 )=0, 2w 2( p, T 0 )0 and w 2( p, T 0 )t0 (see (2.95)) and this is in contradiction with the inequality that is obtained upon substraction of (2.89) and (2.96). (ii) If p # 0 then w 2 =w1 &w satisfies w2( p, T0 )n0=b 2 w 2( p, T 0 ) (see (2.95)) and this is again in contradiction with the inequality that is obtained upon substraction of (2.91) and (2.97). (iii) Finally, if d( p)=\ 1 2 then w 1 ={U } t~  |{ U|  |{U| at (x, t)= ( p, T 0 ) (see (2.94) and (2.95)), and this is in contradiction with (2.90). But, according to the definition of 0 1 , in (2.73), one of the three cases, (i), (ii), or (iii) above, necessarily holds. Then a contradiction has been obtained and the proof is complete. Now we have the ingredients to derive the model (1.17)(1.19). The remainder , in (1.19), is such that |(t)| is appropriately small, and U and V are appropriately close to u and v respectively, as stated in Theorem 1.1.

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93

ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

Proof of Theorem 1.1. If 0d(x)\ 1 2, let '=&d(x) be a coordinate along the outward unit normal to 0, and for each ' 0 let H(' 0 ) be the hypersurface parallel to 0, defined by '=' 0 . Notice that the Laplacian operator may be written as 2U=G &1

U  G +2U. ' '

\

+

(2.98)

Here, for each x # 0 such that d(x)\ 1 2, 2 is the LaplaceBeltrami operator on the hypersurface H(&d(x)), and G= |(1&k 1 ') } } } (1&k m&1 ')|, where k 1 , ..., k m&1 are the principal curvatures of 0 at the point of 0 that shares with x the normal to 0. Notice also that G, G &1,

|G ' |

and

|G '' | are bounded if &\ 1 2'0.

(2.99)

Now, let U=U(x, t) be as defined by (2.60)(2.61), and let U 1 =U 1(', t) be as given by U 1 =S(') &1

|

U(s, t) ds,

where

S(')=

H(')

|

ds

(2.100)

H(')

is the measure of the hypersurface H('). If, for each ' # ]&\ 1 2, 0[ we integrate (2.60) on H(') and divide by S('), then after some manipulations we obtain U 1 t= 2U 1 ' 2 &, 2f (U 1 , V)+. 1(', t)

in

&\ 1 2<'<0,

(2.101)

where we have taken into account that  H(') 2 Uds=0 and

_

. 1(', t)=, 2 f (U 1 , V)& +

1 S(')

|

S"(') 1  U1& S(') 2S(') '

&

f (U, V) ds +

H(')

2S$(') U 1 S(') '

U G ds. H(') G '

|

(2.102)

Similarly, if we integrate the boundary condition (2.60) in 0=H(0) and divide by S 0 =S(0), then we obtain U 1 '=_(1&U 1 )+. 2(t)

at '=0,

(2.103)

where . 2(t)=

1 2S(0)

|

U (G') ds&[S$(0)S(0)] U 1(0, t). H(0) G

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(2.104)

94

MANCEBO AND VEGA

Now, when taking into account assumptions (H.1) and (H.5) and the results in Lemmas 2.1, 2.42.6, and 2.8, we obtain |U 1 | 2[_(,- k 2 2m+_)] exp(&- k 2 2m ,d(x)),

(2.105)

, |U 1(&d(x), t)&U(x, t)| + 3 exp(& - k ,d(x)) |. 1(', t)| ,+ 3 exp(&- k ,')

if x # 0

and

if &\ 1 2'0,

d(x)\ 1 2,

(2.106)

|. 2(t)| + 3 ,

(2.107)

for all tT", where T ", k and k 1 are as defined in Lemma 2.8 and assumption (H.4) and + 3 >0 depends only on the domain 0 and on the quantities (1.20) and satisfies + 3 =O(_(_+,)) in the limit (1.16). Now, let U be the unique solution of Ut= 2U' 2 &, 2f (U, V) U'=_(1&U )

in &<'<0, at

'=0,

(2.108)

if tT", with initial conditions U(', T")=U 1(', T")

if

&\ 1 2'0,

U(', T")=U 1( &\ 1 2, T") exp[- k 1 2m ,('+\ 1 2)]

(2.109)

if &<'< &\ 1 2. Since f(U, V)k 1 U (assumption (H.4)) and U(', T")2[_(,- k2 2m+_)] exp(&- k 2 2m ,d(x)) (see (2.105)), maximum principles readily imply that 0
&<'0

and

tT ",

(2.110)

and assumption (H.5) and the mean function theorem readily imply that f (U 1 , V)& f (U, V)=h(x, t)(U 1 &U ), if

with

h(x, t)k 3

&\ 1 2'0.

(2.111)

Then if *=- k2

and

$=max[8+ 3 k,, 4+ 3 - k ,,

[4_(_+,- k 2 2m)] exp(&- k ,\ 1 4)],

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95

ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

the functions w \ =$ exp(*,')\(U 1 &U ) are readily seen to satisfy w \ t& 2w \ ' 2 +, 2h(x, t) w \ >0 in &\ 1 2'0, w \ >0 w \ >0

at '=&\ 1 2, in

w \ '+_w \ >0

&\ 1 2'0,

if

if tT ",

at '=0,

if

tT ",

t=T ",

and maximum principles readily imply that w + 0, i.e., that |U 1 &U | $ exp(*,')

in

&\ 1 2'0,

if

tT ".

(2.112)

If, in addition, we take into account (2.64) and (2.106), then (1.23) follows. In order to obtain (1.24) we only need to take into account that V satisfies (2.62), and that u, v&V and u&U satisfy (2.12), (2.33) and (1.23). Thus the proof is complete. Remark 2.10. In the following, we shall ignore the initial transient 0tT, where T is as given in Theorem 1.1. Then the estimates (1.23)(1.24) and (2.110) will be assumed to hold for all {0.

3. ANALYSIS OF THE ASYMPTOTIC MODEL The asymptotic model (1.17)(1.19) will be considered now. We shall first analyze, in Section 3.1, the distinguished limit when all terms are of the same order and then we shall consider, in Section 3.2, other sub-limits leading to still simpler sub-models. Finally, in Section 3.3 we shall analyze the particular case when the non-linearity is as given by one of the expressions (1.4)(1.6), and the activation energy # is large. 3.1. The Distinguished Limit Let us consider the following sub-limit of (1.16) _,  s,

&L, 2  V 0 lS 0 ,

;,&  *,

(3.1)

where s>0,

l>0

and

*>0 are bounded, and ,  ,

L  ,

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&  0. (3.2)

96

MANCEBO AND VEGA

Then the model (1.17)(1.19) may be written as U{= 2U! 2 & f (U, V) U =0

at !=&,

l &1 dVd{=1&V+*

|

in &
(3.3) at !=0,

(3.4)

0

f (U, V) d!+ 1({),

(3.5)

{=, 2t,

(3.6)

&

where !=,',

 1 =&

and, according to Remark 2.10, 0
if &
uniformly in 0{<.

and

{0, (3.7) (3.8)

Two remarks concerning this model are in order: (a) According to Theorem 1.1, the attractors as {   of (3.3)(3.5) are close to the attractors of the original model (1.1)(1.2), in the sense of the estimates (1.23)(1.24). (b)

If we ignore the remainder  1 then (3.5) may be rewritten as l &1 dVd{=1&V+*

|

0

f (U, V) d!.

(3.5$)

&

Notice that condition (3.7) defines an invariant set of both (3.3)(3.5) and (3.3)(3.4), (3.5$) (that is, if the first two inequalities in (3.7) hold at {={ 0 , then they also hold for all {{ 0 ), as readily seen when applying a maximum principle. Then we may consider only those solutions of both (3.3)(3.5) and (3.3)(3.4), (3.5) that satisfy (3.7) and (for comparison of the solutions of both problems) define the distance associated with the norm &(U( } , {), V({))&=sup[U(!, {) exp(&- k 2 2m !) : &
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ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

97

(3.3)(3.5) and (3.3)(3.4), (3.5$) are close to each other; of course, nonexponential attractors need not be close. This is the sense in which the asymptotic behavior as {   of (3.3)(3.5) (or that of (1.1)(1.3), according to Remark (a) above) may be approximated by that of (3.3)(3.4), (3.5'). Even with a fairly simple nonlinearity, such as that in (1.4), with p=1, the model (3.3)(3.4), (3.5$) exhibits at least multiple steady states and Hopf bifurcations, see [2]. 3.2. Some Particular Sub-limits of (3.1)(3.2) Let us now consider the model (3.3)(3.5) in the particular sub-limits s  0, s   and l  0. As s0

(3.10)

U is small (see (3.7)) and, according to assumption (H.5), at the end of Section 1, the nonlinearity f may be written as f (U, V)= f u(0, V) U +O( |U | 2 ).

(3.11)

If * is fixed then the right-hand side of (3.5) equals 1&V in first approximation and the dynamics of the resulting model is trivial. If, instead, * is large, such that s*  * 1 {0, ,

with

0{l=fixed,

(3.12)

then the model (3.3)(3.5) may be rewritten as U 1 {= 2U 1 ! 2 &U 1 f 1(V)+ 2(!, {) exp(- k 2 2m !)

in &
U 1 =0

at !=&,

l &1 dVd{=1&V+* 1 f 1(V)

U 1 !=1+ 3({)

|

at

!=0,

(3.14)

0

U 1 d!+ 1({)+ 4({),

(3.15)

f 1(V)= f u(0, V)

(3.16)

&

where U 1 =Us,

and, according to (3.7) and (3.11), the remainders  2 ,  3 and  4 are such that | 2(!, {)| + | 3({)| + | 4({)| =O(s)

uniformly in &
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{0 (3.17)

98

MANCEBO AND VEGA

in the limit (3.10), (3.12). Then, if the remainders  1 , ...,  5 are ignored in (3.13)(3.15), we obtain an asymptotic model that is seen to approximate the large time behavior in a sense similar to that described in Remarks (a) and (b), at the end of Section 3.1. In the limit s  ,

with

l{0

and

*{0

fixed,

(3.18)

we have |U(0, {)!| =uniformly bounded in {0,

(3.19)

as readily seen by an argument similar to that in the proof of Lemma 2.6, Case 1. Then the boundary conditions (3.4) may be written as U =0

U =1+ 5({)

at !=&,

at !=0,

(3.4$)

where the remainder  5 is given by  5({)=s &1 U(0, {)!

(3.20)

and, according to (3.19), satisfies | 5({)| =O(s &1 )

uniformly in

{0,

(3.21)

in the limit (3.18). Again, if the remainders  1 and  5 are ignored in the model (3.3), (3.4$), (3.5), then we obtain an asymptotic model that approximates the large time behavior of (1.1)(1.2), in a sense similar to that described in Remarks (a) and (b), at the end of Section 3.1. In the limit. l  0,

with s and * fixed,

(3.22)

let us replace Eq. (3.3) by 0= 2U! 2 & f (U, V)

in &
(3.3$)

For each function V=V({) the problem (3.3$) (3.4) is readily seen to uniquely define U =H(!, V)

(3.23)

and, according to assumptions (H.4) and (H.5) at the end of Section 1, the following estimates are seen to hold 0
|HV| exp(&- k !)=O(1)

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(3.24)

99

ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

uniformly in &0, V>12, in the limit (3.22), where O(1) stands for a bounded quantity. If (U, V) is a solution of (3.3)(3.5) and U 1 =H(!, V({)) is the associated solution of (3.3$), (3.4), then |dVd{| = O(l ) uniformly in {>0, as readily seen from (3.5) when taking into account assumption (H.4) and Eq. (3.7), and (3.24) implies that |U 1 {| exp(&- k 2 !)=O(l )

uniformly in &0, (3.25)

in the limit (3.22). Then we only need to apply maximum principles, as in the argument leading to Eq. (2.108), in the proof of Theorem 1.1, to obtain |U &U 1 | =O(1) exp(&k{+- k !)+O(l ) exp(- k !) uniformly in &0, where k=min[k 2 , k 2 ]2 (see assumptions (H.4) and (H.5)), and consequently |U &U 1 | exp( &- k !)=O(l ) uniformly in &
{>{ 0 =O( |log l| ).

(3.26)

Then we only need to replace U by U 1 in (3.5) to obtain dVd{~ =1&V+*

|

0

f (H(!, V), V) d!+ 1({)+ 6({),

(3.27)

&

where H is as given in (3.23); the new time variable {~ and the remainder  6 are given by {~ =lt,

 6({)=*

|

0

[ f (U, V)& f (U 1 , V)] d! &

and, according to (3.26) the remainder is small, i.e., | 6({)| =O(l)

uniformly in

&{ 0 ,

(3.28)

in the limit (3.22). If the remainders  1 and  6 are ignored in (3.27) then an autonomous ODE is obtained that may exhibit multiple steady states and yields trivial dynamics (namely, V({) converges to a steady state as {  ). As above, this implies that the dynamics of (1.1)(1.2) is essentially trivial in first approximation. Similarly, as l   one could try to prove that the time derivative in the left hand side of (3.5) may be just omitted in first approximation. Then, after solving the resulting equation and replacing its solutions into (3.3), a non-local semilinear equation would be obtained. Unfortunately, this

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100

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would require the non-linearity f to be such that f v <0, while f v is usually positive (see (1.4)(1.6)). In this case, when the time derivative is omitted in (3.5), the resulting equation may possess multiple solutions and, as a consequence, the complete problem may possess relaxation oscillations whose analysis is beyond the scope of this paper (see, e.g., Hastings [16] and Grasman [17] for a formal analysis of these oscillations in related problems). 3.3. Large Activation Energies As mentioned in Section 1, the activation energy # may be fairly large (1.4)(1.6). Let us now consider the limit #

(3.29)

in (1.4) (with p=1 for assumptions (H.4) and (H.5) to hold); the analysis of the non-linearities (1.5) and (1.6) is completely similar. In the limit (3.29) we shall consider two distinguished limits. In a first extinction limit we rescale , 2, ;, and v as , 21 =, 2 exp(&#),

; 1 =;#

and

v 1 =v#,

(3.30)

to rewrite (1.1)(1.2), (1.4) as ut=2u&, 21 u exp(&1v 1 ) un=_(1&u)

in 0,

at 0,

(3.31)

L &1v 1 t=2v 1 +; 1 , 12u exp( &1v 1 ) v 1 n=&(1#&v 1 )

in 0,

at 0.

(3.32)

In the limit #  ,

, 21  ,

; 1 , 1 _(, 1 +_)  0,

&  0,

_ &1 =O(1)

(3.33)

the results in Section 2 (and, in particular, Theorem 1.1) apply to yield the asymptotic model Ut= 2U' 2 &, 21 U exp(&1V 1 ) U =0

at '=&,

in &<'<0,

U'=_(1&U )

(V 0 S 0 L) dV 1 dt=&&V 1 +; 1 , 21 exp(&1V 1 )

at

|

'=0,

(3.34) (3.35)

0 &

U d!+&#+(t), (3.36)

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ASYMPTOTIC JUSTIFICATION OF A 1-D MODEL

101

where V 1 is the spatial average of v 1 and |(t)+&#|, |u&U | and |v 1 &V 1 | are appropriately small, according to Theorem 1.1. Notice that (3.34)(3.36) is essentially (except for the constant & on the right-hand side of (1.19)) a particular case of the model (1.17)(1.19), and thus the analysis in Sections 3.13.2 above applies to this new asymptotic model. In a second ignition limit, ; and v&1 are rescaled as ; 2 =#;

and

v 2 =#(v&1),

(3.37)

to rewrite (1.1)(1.2), (1.4) as ut=2u&, 2u exp[v 2 (1+v 2 #)] un=_(1&u)

in 0,

at 0,

(3.38)

L &1v 2 t=2v 2 +; 2 , 2u exp[v 2 (1+v 2 #)] v 2 n=&&v 2

in 0,

at 0.

(3.39)

Notice that now the non-linearity is not bounded (as v 2  ) in the limit (3.29), as required by assumption (H.3) and thus the results in Section 2 do not apply to (3.38)(3.39). But the boundedness assumption was used in Section 2 only to prove (in Lemma 2.2) that v is bounded. Then, if we only consider those solutions of (3.38)(3.39) such that |v 2(t)| is bounded in 0
, 2  ,

; 2 ,_(,+_)  0,

&  0,

_ &1 =O(1),

(3.40)

to obtain the asymptotic model Ut= 2U' 2 &, 2U exp(V 2 ) U =0

at

'=&,

in

&<'<0,

U'=_(1&U )

at '=0,

0 (V 0 S 0 L) dV 2 dt=&&V 2 +; 2 , 2 exp(V 2 )  & U d'+(t),

(3.41) (3.42) (3.43)

where V 2 is the spatial average of v 2 and ||, |u&U | and |v 2 &V 2 | are appropriately small. Again, the analysis in Section 3.1 and 3.2 above still applies to (3.41)(3.43). In addition, (3.38)(3.39) possess solutions that are not bounded, but become very large in finite time. This phenomenon is known as ignition in the Combustion literature and its analysis in connection with (3.38)(3.39) is (again) beyond the scope of this paper.

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4. CONCLUDING REMARKS We have considered the model (1.1)(1.2) in the limit (1.16). The spatial domain, the non-linearity and the initial data have been assumed to satisfy assumptions (H.1)(H.6). Some of these assumptions could be relaxed as explained at the end of Section 1, and have been imposed for the sake of both brevity and clarity. The assumption f u >k 3 >0 (in (H.5)) instead, is necessary for some of the ideas in the paper to apply, but perhaps it is not necessary for the main result to hold. In Section 2 we have first obtained some estimates on the solutions of (1.1)(1.2) implying that, after an initial transient (i) the reactant concentration u becomes quite small except in a boundary layer, near the boundary of the domain, 0, (ii) the temperature v becomes approximately spatially constant, and (iii) the gradient of u along the hypersurfaces parallel to 0 becomes small. Then the asymptotic model (1.17)(1.19) was obtained. The 1-D parabolic semilinear equation (1.17) yields the reactant concentration in the above-mentioned boundary layer, and the ODE (1.19) gives the spatial average of the temperature. The asymptotic model was analyzed (in Section 3.1) in the distinguished limit when all terms are comparable (except for the remainder , that is smaller) and in some representative sub-limits (in Section 3.2). Namely, (i) when s  0 (i.e., when chemical reaction is much faster than material exchange through the boundary) and s   the mixed boundary condition at '=0 (in (1.18)) can be replaced by Neumann and Dirichlet boundary conditions respectively, and (ii) when l  0 (i.e., when diffusion is much faster that thermal exchange through the boundary) the reactant concentration becomes quasi-steady and the asymptotic model is reduced to an ODE; the opposite limit, l  , is much more subtle, as explained at the end of Section 3.2. Finally, in Section 3.3 we considered the case when the chemical reaction obeys a first-order Arrhenius kinetic law and the activation energy is large. Let us point out that the asymptotic model was derived in a quite realistic limit, and that it is much simpler than the original reaction diffusion system. Thus we expect this model to be useful in the analysis of the dynamics of catalytic pellets, which are of great interest in chemical reactor theory.

REFERENCES 1 R. Aris, ``The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts,'' Vols. I and II, Clarendon Press, Oxford, 1975. 2. I. E. Parra and J. M. Vega, Local non-linear stability of the steady state in an isothermal catalysts, SIAM J. Appl. Math. 48 (1988), 854881.

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3. F. J. Mancebo and J. M. Vega, An asymptotic justification of a model of isothermal catalysts, J. Math. Anal. Appl. 175 (1993), 523536. 4. E. Conway, D. Hoff, and J. Smoller, Large time behavior of solutions of nonlinear reactiondiffusion equations, SIAM J. Appl. Math. 35 (1978), 116. 5. J. K. Hale and C. Rocha, Varying boundary conditions with large diffusivity, J. Math. Pures Appl. 66 (1987), 139158. 6. J. K. Hale and C. Rocha, Interaction of diffusion and boundary conditions, Nonlinear Anal. 11 (1987), 633649. 7. J. K. Hale and K. Sakamoto, Shadow systems and attractors in reaction-diffusion equations, Appl. Anal. 32 (1989), 287303. 8. J. M. Vega, Invariant regions and global asymptotic stability in an isothermal catalyst, SIAM J. Math. Anal. 19 (1988), 774796. 9. A. di Liddo and L. Maddalena, Mathematical analysis of a chemical reaction with lumped temperature and strong absorption, J. Math. Anal. Appl. 163 (1992), 86103. 10. F. J. Mancebo and J. M. Vega, Weakly non-uniform thermal effects in a porous catalyst: Asymptotic models and local nonlinear stability of the steady states, SIAM J. Appl. Math. 52 (1992), 12381259. 11. F. J. Mancebo and J. M. Vega, A model of porous catalyst accounting for incipiently nonisothermal effects, in preparation, 1996. 12. M. H. Protter and H. Weinberger, ``Maximum Principles in Differential Equations,'' PrenticeHall, Englewood Cliffs, NJ, 1967 13. O. A. Ladyz henskaja and N. N. Uralceva, Linear and quasilinear elliptic equations, Academic Press, New York, 1968. 14. O. A. Ladyz henskaja, V. A. Solonnikov, and N. N. Uralceva, ``Linear and Quasilinear Equations of Parabolic Type,'' Amer. Math. Soc., Providence, RI, 1968. 15. H. W. Guggenheimer, ``Differential Geometry,'' Dover, New York, 1977. 16. S. P. Hastings, Formal relaxation oscillations for a model of a catalytic particle, Quat. Appl. Math. 41 (1984), 395405. 17. J. Grasman, ``Asymptotic Methods for Relaxation Oscillations and Applications,'' Springer-Verlag, BerlinNew York, 1987.

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